(9x^3xy^2-2xy^2 +6x^2.y) . (-3xy)

To simplify the given expression \((9x^3y^2 - 2xy^2 + 6x^2y)(-3xy)(9x^3y^2 - 2xy^2 + 6x^2y)(-3xy)\), we can proceed step by step.

### Step 1: Simplifying the Expression

First, let's rewrite the expression:

\[
(9x^3y^2 - 2xy^2 + 6x^2y)(-3xy)(9x^3y^2 - 2xy^2 + 6x^2y)(-3xy)
\]

Notice that we have the term \((9x^3y^2 - 2xy^2 + 6x^2y)\) appearing twice. We can denote it as \(A\) for convenience:

\[
A = 9x^3y^2 - 2xy^2 + 6x^2y
\]

Then the expression simplifies to:

\[
A \cdot (-3xy) \cdot A \cdot (-3xy)
\]

Which can be rewritten as:

\[
A^2 \cdot (-3xy)^2
\]

### Step 2: Calculating \((-3xy)^2\)

Now, calculate \((-3xy)^2\):

\[
(-3xy)^2 = 9x^2y^2
\]

### Step 3: Now, Calculate \(A^2\)

Now we will expand \(A^2\):

\[
A^2 = (9x^3y^2 - 2xy^2 + 6x^2y)(9x^3y^2 - 2xy^2 + 6x^2y)
\]

For this expansion, we will use the distributive property (or FOIL method for binomials):

1. **First Terms**: \(9x^3y^2 \cdot 9x^3y^2 = 81x^6y^4\)
2. **Outer Terms**: \(9x^3y^2 \cdot (-2xy^2) = -18x^4y^4\)
3. **Inner Terms**: \((-2xy^2) \cdot 9x^3y^2 = -18x^4y^4\)
4. **Last Terms**: \((-2xy^2) \cdot (-2xy^2) = 4x^2y^4\)
5. **Cross Terms** (with \(6x^2y\)):
- \(9x^3y^2 \cdot 6x^2y = 54x^5y^3\)
- \(6x^2y \cdot 9x^3y^2 = 54x^5y^3\)
- \(6x^2y \cdot (-2xy^2) = -12x^3y^4\)
- \((-2xy^2) \cdot 6x^2y = -12x^3y^4\)
- \(6x^2y \cdot 6x^2y = 36x^4y^2\)

### Collecting Like Terms

Now, let's gather all these:

\[
A^2 = 81x^6y^4 + (-18x^4y^4 - 18x^4y^4) + (54x^5y^3 + 54x^5y^3) + (-12x^3y^4 - 12x^3y^4) + 36x^4y^2 + 4x^2y^4
\]
\[
= 81x^6y^4 - 36x^4y^4 + 108x^5y^3 - 24x^3y^4 + 36x^4y^2 + 4x^2y^4
\]

This can be combined into one polynomial:

\[
A^2 = 81x^6y^4 + 108x^5y^3 - 36x^4y^4 + 36x^4y^2 - 24x^3y^4 + 4x^2y^4
\]

### Final Expression

Now, we can multiply the result \(A^2\) by \(9x^2y^2\):

\[
9x^2y^2(81x^6y^4 + 108x^5y^3 - 36x^4y^4 + 36x^4y^2 - 24x^3y^4 + 4x^2y^4)
\]

Distributing \(9x^2y^2\) gives:

\[
729x^8y^6 + 972x^7y^5 - 324x^6y^6 + 324x^6y^4 - 216x^5y^6 + 36x^4y^6
\]

### Conclusion

The complete simplified expression is:

\[
729x^8y^6 + 972x^7y^5 - 324x^6y^6 + 324x^6y^4 - 216x^5y^6 + 36x^4y^6
\]

Feel free to ask if you have further questions!